Extremal orders of the Zeckendorf sum of digits of powers

TL;DR

This paper investigates the extremal growth rates of the Zeckendorf sum of digits function for powers of integers, revealing bounds and existence results that extend classical Fibonacci power problems.

Contribution

It establishes extremal orders of the Zeckendorf sum of digits for powers and provides bounds on the number of integers with specific digit sum ratios, extending prior Fibonacci power research.

Findings

Derived extremal minimal and maximal orders of s_F(n^h)/s_F(n).

Proved the existence of n with controlled sum of Fibonacci numbers for powers.

Provided bounds on the frequency of small and large ratios of s_F(n^h)/s_F(n).

Abstract

Denote by s_F(n) the minimal number of Fibonacci numbers needed to write n as a sum of Fibonacci numbers. We obtain the extremal minimal and maximal orders of magnitude of s_F(n^h)/s_F(n) for any h>= 2. We use this to show that for all $>N_0(h) there is an n such that n is the sum of N Fibonacci numbers and n^h is the sum of at most 130 h^2 Fibonacci numbers. Moreover, we give upper and lower bounds on the number of n's with small and large values of s_F(n^h)/s_F(n). This extends a problem of Stolarsky to the Zeckendorf representation of powers, and it is in line with the classical investigation of finding perfect powers among the Fibonacci numbers and their finite sums.